Non-Commutative Mechanics in Mathematical & in Condensed Matter Physics

@article{Horvathy2006NonCommutativeMI,
  title={Non-Commutative Mechanics in Mathematical \& in Condensed Matter Physics},
  author={P. A. Horvathy},
  journal={Symmetry Integrability and Geometry-methods and Applications},
  year={2006},
  volume={2},
  pages={090}
}
  • P. Horvathy
  • Published 22 September 2006
  • Physics
  • Symmetry Integrability and Geometry-methods and Applications
Non-commutative structures were introduced, independently and around the same time, in mathematical and in condensed matter physics (see Table 1). Souriau’s construction applied to the two-parameter central extension of the planar Galilei group leads to the “exotic” particle, which has non-commuting position coordinates. A Berryphase argument applied to the Bloch electron yields in turn a semiclassical model that has been used to explain the anomalous/spin/optical Hall effects. The non… 

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References

SHOWING 1-10 OF 50 REFERENCES

monopole and Berry phase in momentum space in noncommutative quantum mechanics

To build genuine generators of the rotations group in noncommutative quantum mechanics, we show that it is necessary to extend the noncommutative parameter $\ensuremath{\theta}$ to a field operator,

Exotic Galilean symmetry in the non-commutative plane and the Hall effect

Quantum mechanics in the non-commutative plane is shown to admit the 'exotic' symmetry of the doubly centrally extended Galilei group. When coupled to a planar magnetic field whose strength is the

The Non-commutative Landau Problem

Abstract The Landau problem is discussed in two similar but still different non-commutative frameworks. The “standard” one, where the coupling to the gauge field is achieved using Poisson brackets,

"Topological" (Chern-Simons) quantum mechanics.

Two quantum-mechanical models are constructed that are analogs of three-dimensional, topologically massive as well as Chern-Simons gauge-field theories, and the phase-space reductive limiting procedure that takes the former to the latter is studied.

2 0 Se p 20 06 Anomalous Hall Effect in non-commutative mechanics

The anomalous velocity term in the semiclassical model of a Bloch electron deviates the trajectory from the conventional one. When the Berry curvature (alias noncommutative parameter) is a monopole

Noncommuting coordinates in the Hall effect and in vortex dynamics

Laughlin's Ansatz to explain the fractional Quantum Hall effect is derived by coupling a particle associated with ``exotic'' the two-fold central extension of the planar Galilei group. The reduced

Noncommuting Coordinates, Exotic Particles, and Anomalous Anyons in the Hall Effect

We review our previous “exotic” particle, together with the more recent anomalous anyon model (which has the arbitrary gyromagnetic factor g). The nonrelativistic limit of the anyon generalizes the